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A058398 Partition triangle A008284 read from right to left.

%I A058398 #27 Apr 24 2024 13:15:45
%S A058398 1,1,1,1,1,1,1,1,2,1,1,1,2,2,1,1,1,2,3,3,1,1,1,2,3,4,3,1,1,1,2,3,5,5,
%T A058398 4,1,1,1,2,3,5,6,7,4,1,1,1,2,3,5,7,9,8,5,1,1,1,2,3,5,7,10,11,10,5,1,1,
%U A058398 1,2,3,5,7,11,13,15,12,6,1,1,1,2,3,5,7,11,14,18,18,14,6,1,1,1,2,3,5,7,11
%N A058398 Partition triangle A008284 read from right to left.
%C A058398 a(n,m) is the number of partitions of n with n-(m-1) parts or, equivalently, with greatest part n-(m-1).
%C A058398 The columns are the diagonals of triangle A008284. The diagonals are the columns of the partition array p(n,m), n >= 0, m >= 1, with p(n,m) the number of partitions of n in which every part is <= m; p(0,m) := 1. For n >= 1 this array is obtained from table A026820 read as lower triangular array with extension of the rows according to p(n,m)=A000041(n) for m>n.
%D A058398 L. Comtet, Advanced Combinatorics, Reidel, 1974, pp. 94, 96 and 307.
%D A058398 M. Kauers and P. Paule, The Concrete Tetrahedron, Springer 2011, p. 27.
%H A058398 Seiichi Manyama, <a href="/A058398/b058398.txt">Rows n = 1..140, flattened</a>
%H A058398 Henry Bottomley, <a href="/A008284/a008284.gif">Illustration of initial terms</a>
%H A058398 Roser Homs and Anna-Lena Winz, <a href="https://arxiv.org/abs/2309.06871">Deformations of local Artin rings via Hilbert-Burch matrices</a>, arXiv:2309.06871 [math.AC], 2023. See p. 16.
%F A058398 a(n, m)= A008284(n, n-(m-1)).
%F A058398 a(n, m)= p(m-1, n-m+1), n >= m >= 1 with the p(n, m) array defined in the comment.
%F A058398 a(n, m)=0 if n<m or m<=0 or n=0; a(1, 1)=1; a(n, m)= a(n-1, m)+a(m-1, 2*m-n+1).
%F A058398 Viewed as a square array by antidiagonals, T(n,k) = 0 if n<0; T(n,1) = 1; otherwise T(n,k) = T(n,k-1) + T(n-k,k). - _Franklin T. Adams-Watters_, Jul 25 2006
%F A058398 Let x be a triangular number C(n,2), where n is the integer being partitioned.  Then a(x) = a(x+1) = a(x+2) = 1.  Also, a(x+3) = 2 for x>3 and a(x-1) = floor(n/2). - _Allan Bickle_, Apr 18 2024
%e A058398 Lower triangular matrix:
%e A058398   1;
%e A058398   1,1;
%e A058398   1,1,1;
%e A058398   1,1,2,1;
%e A058398   1,1,2,2,1;
%e A058398   1,1,2,3,3,1;
%e A058398   1,1,2,3,4,3,1;
%e A058398   1,1,2,3,5,5,4,1;
%e A058398   ...
%t A058398 row[n_] := Table[ IntegerPartitions[n, k] // Length, {k, 0, n}] // Differences // Reverse; Table[row[n], {n, 1, 14}] // Flatten (* _Jean-François Alcover_, Feb 28 2013 *)
%Y A058398 Cf. A026820, A008284, A000041.
%K A058398 nonn,easy,tabl,nice
%O A058398 1,9
%A A058398 _Wolfdieter Lang_, Dec 11 2000